Bernstein–Sato polynomial

- If f(x)=x_1^{n_1}x_2^{n_2}\cdots x_r^{n_r} then ::\prod_{j=1}^r\partial_{x_j}^{n_j}\quad f(x)^{s+1} =\prod_{j=1}^r\prod_{i=1}^{n_j}(n_js+i)\quad f(x)^s :so ::b(s)=\prod_{j=1}^r\prod_{i=1}^{n_j}\left(s+\frac{i}{n_j}\right). - The Bernstein–Sato polynomial of *x*2 + *y*3 is ::(s+1)\left(s+\frac{5}{6}\right)\left(s+\frac{7}{6}\right). - If *t**ij* are *n*2 variables, then the Bernstein–Sato polynomial of det(*t**ij*) is given by ::(s+1)(s+2)\cdots(s+n) :which follows from ::\Omega(\det(t_{ij})^s) = s(s+1)\cdots(s+n-1)\det(t_{ij})^{s-1} :where Ω is Cayley's omega process, which in turn follows from the Capelli identity. ## Applications - If f(x) is a non-negative polynomial then f(x)^s, initially defined for *s* with non-negative real part, can be analytically continued to a meromorphic distribution-valued function of *s* by repeatedly using the functional equation ::f(x)^s={1\over b(s)} P(s)f(x)^{s+1}. :It may have poles whenever *b*(*s* + *n*) is zero for a non-negative integer *n*. - If *f*(*x*) is a polynomial, not identically zero, then it has an inverse *g* that is a distribution;Warning: The inverse is not unique in general, because if *f* has zeros then there are distributions whose product with *f* is zero, and adding one of these to an inverse of *f* is another inverse of *f*. in other words, *f g* = 1 as distributions. If *f*(*x*) is non-negative the inverse can be constructed using the Bernstein–Sato polynomial by taking the constant term of the Laurent expansion of *f*(*x*)*s* at *s* = −1. For arbitrary *f*(*x*) just take \bar f(x) times the inverse of \bar f(x)f(x). - The Malgrange–Ehrenpreis theorem states that every differential operator with constant coefficients has a Green's function. By taking Fourier transforms this follows from the fact that every polynomial has a distributional inverse, which is proved in the paragraph above. - Pavel Etingof (1999) showed how to use the Bernstein polynomial to define dimensional regularization rigorously, in the massive Euclidean case. - The Bernstein-Sato functional equation is used in computations of some of the more complex kinds of singular integrals occurring in quantum field theory. Such computations are needed for precision measurements in elementary particle physics as practiced for instance at CERN (see the papers citing). However, the most interesting cases require a simple generalization of the Bernstein-Sato functional equation to the product of two polynomials (f_1(x))^{s_1}(f_2(x))^{s_2}, with *x* having 2-6 scalar components, and the pair of polynomials having orders 2 and 3. Unfortunately, a brute force determination of the corresponding differential operators P(s_1,s_2) and b(s_1,s_2) for such cases has so far proved prohibitively cumbersome. Devising ways to bypass the combinatorial explosion of the brute force algorithm would be of great value in such applications. ## Notes ## References ## References 1. Bernshtein, I. N.. (1971). ["Modules over a ring of differential operators. Study of the fundamental solutions of equations with constant coefficients"](http://link.springer.com/10.1007/BF01076413). *Functional Analysis and Its Applications*. 2. (June 1972). "On Zeta Functions Associated with Prehomogeneous Vector Spaces". *Proceedings of the National Academy of Sciences*. 3. (July 1974). ["On Zeta Functions Associated with Prehomogeneous Vector Spaces"](https://www.jstor.org/stable/1970844). *The Annals of Mathematics*. 4. (December 1990). ["Theory of prehomogeneous vector spaces (algebraic part)—the English translation of Sato's lecture from Shintani's note"](https://www.cambridge.org/core/product/identifier/S0027763000003214/type/journal_article). *Nagoya Mathematical Journal*. 5. Coutinho, Severino C.. (1995). "A primer of algebraic D-modules". *[[Cambridge University Press]]*. 6. Borel, Armand. (1987). "Algebraic D-Modules". *[[Academic Press]]*. 7. Kashiwara, Masaki. (2003). "D-modules and microlocal calculus". *[[American Mathematical Society]]*. 8. Kashiwara, Masaki. (February 1976). ["B-functions and holonomic systems: Rationality of roots ofB-functions"](http://link.springer.com/10.1007/BF01390168). *Inventiones Mathematicae*. 9. Sabbah, C.. (1987). ["Proximité évanescente. I. La structure polaire d'un $\mathcal {D}$-module"](https://www.numdam.org/item/?id=CM_1987__62_3_283_0). *Compositio Mathematica*. 10. (May 2006). ["Bernstein–Sato polynomials of arbitrary varieties"](http://www.journals.cambridge.org/abstract_S0010437X06002193). *Compositio Mathematica*. 11. (June 28, 2009). "Proceedings of the 2009 international symposium on Symbolic and algebraic computation". *ACM*. 12. (2010-07-25). "Proceedings of the 2010 International Symposium on Symbolic and Algebraic Computation". *ACM*. 13. (2000). "Quantum fields and strings: a course for mathematicians. Vol. 1". *American Mathematical Society [u.a.]*. 14. Tkachov, Fyodor V. (April 1997). ["Algebraic algorithms for multiloop calculations The first 15 years. What's next?"](https://linkinghub.elsevier.com/retrieve/pii/S0168900297001101). *Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment*. ::callout[type=info title="Wikipedia Source"] This article was imported from [Wikipedia](https://en.wikipedia.org/wiki/Bernstein–Sato_polynomial) and is available under the [Creative Commons Attribution-ShareAlike 4.0 License](https://creativecommons.org/licenses/by-sa/4.0/). Content has been adapted to SurfDoc format. Original contributors can be found on the [article history page](https://en.wikipedia.org/wiki/Bernstein–Sato_polynomial?action=history). ::